I am struggling with coming up with a few examples for an upcoming exam. I do not need justification for your example, I would rather figure that out for myself! Here are the examples I need help with:
1) A power series $f(x) = \sum_{n=0}^{\infty} a_nx^n$ that converges on $[-1,1]$ but whose term-by-term derivative $f'(x)=\sum_{n=0}^{\infty} na_nx^{n-1}$ fails to converge at one or both endpoints.
2) A compact set $K \subset \mathbb{R}$ and a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the inverse image $f^{-1}(K)$ is not compact.
3) Functions $f_n, n=1,2,...$ such that $f_n \rightarrow f$ pointwise on $[0,1]$ but $\lim_{n\rightarrow \infty} \int_0^1 f_n \neq \int_0^1 f$
4) A function $f: \mathbb{R} \rightarrow \mathbb{R}$ differentiable at $0$, with $f'(0)\neq 0 $, and $f$ not invertible in a neighborhood of the point $0$.
Again, all I need is an example for each. I would like to come up with the justification myself.
Thanks!
$(1)$ Try $f(x)=arctan (x)$, who is defined on $[-1,1]$ and whose Taylor series converges on this interval. Note the derivative of $f$ is discontinuous at $\pm i$. What does that say about radius of convergence in $\mathbb{C}$?
$(2)$ You can show any singleton $\{ a : a\in\mathbb{R} \}$ is a closed set. So consider the constant function $f(x)=a$ for all $x\in\mathbb{R}$
$(3)$ Consider the sequence $( f_n )_{n=1}^{\infty}$ where $$f_n (x) = \begin{cases} n & \mbox{ if }x \in [0, 1/n) \\ 0 & \mbox{ otherwise }\end{cases}$$
$(4)$ I don't think there is an example. The Inverse Function Theorem states, among other things, that "Given a continuous and differentiable function $f$ with non-zero derivative at a point $x_0$,then $f$ is invertible on a neighbourhood of $x_0$"
I think what you meant to ask for $(4)$ is "Give an example of a function $f$ such that $f'(0)=0$ but $f$ is not invertible on any neighbourhood of $0$". Then in this case, you could use the function $f(x)=x^2$